University of Nova Gorica, Vipavska 13, SI-5000 Nova Gorica, Slovenia. 1. INTRODUCTION. Several stochastic models [8] for modeling the microstructure ...
Point automata method for prediction of grain structure in the continuous casting of steel A.Z. Lorbiecka1, B. Šarler2 1
University of Nova Gorica, Vipavska 13, SI-5000 Nova Gorica, Slovenia
ABSTRACT: This paper represents a new point automata (PA) approach, developed for the simulation of solidification grain structure formation (equiaxed to columnar (ECT) and columnar to equiaxed (CET) transition) during the continuous casting (CC) process of steel billets. The physical model is composed of the macroscopic heat transfer model of the CC [10,11] process and microscopic model, composed of Gaussian nucleation and KurtzGiovanola-Trivedi (KGT) [15] growth parts. In our previous paper [7] the numerical solution of microstructure was based on the conventional Cellular Automata (CA) method. The microscopic model parameters have been adjusted with respect to the experimental Baumann prints. In the present work, the CA cells have been replaced by a new, random point based approach, first proposed by Janssens [7] for recrystallisation. The differences in numerical implementation of the classical CA and the new PA microstucture solution procedure are compared and discussed. The main advantage of the new approach represents the possibility of flexible density of the nodes and flexible definition of neighborhood configuration. 1.
INTRODUCTION
Several stochastic models [8] for modeling the microstructure formation have been developed and applied over the past years. The main goal of this work is to develop a new numerical approach for solving the microstructure evolution and to implement it into simulation tool for modeling the grain structure during the CC solidification process. There is a growing interest in computational modeling of CC of steel in order to be able to predict the properties of the product. This paper represents an extension of our previous work [7], where the macro-micro numerical simulator for modeling the microstructure was presented and validated with the experimental data for spring steel 51CrMoV4. The macroscopic heat transfer model was solved by a meshless method and the microscopic grain growth model was solved by the classical CA method. Here, for the first time, the numerical solution of related microstructure equations is recalculated by using a new irregular CA method. Irregular CA method or PA method was first proposed by Thieme-Marti [13] and then by Flache and Hegselmann [1]. In the present paper we follow the Janssens approach [2], where the irregular CA method was applied to model the recrystallisation process [3, 4]. This approach is in the present paper adapted to simulation of the ECT and CET transformations. An elaboration of our previous microstructure modeling, based on conventional CA method, can be found in [6,7]. MICRO CELLS
MACRO MESHLESS NODES
MICRO POINTS
Fig. 1: Center: Example of meshless macroscopic discretisation with 28 x 28 macro nodes, left: conventional CA micro model discretisation with 25 x 25 micro CA cells, right: random PA micro model discretisation with 167 points
2.
PHYSICAL MODEL OF MICROSTRUCTURE FORMATION
The macroscopic model for calculation of temperature field in CC steel billet is elaborated in [7], together with the coupling between the macroscopic and the microscopic calculations [14]. Exactly the same model and coupling are used in the present paper. 2.1
Nucleation model
A continuous nucleation model is adopted in the present work, based on the Gaussian distribution [5], in which two different nucleation areas are considered: at the surface (s) and in the bulk (b) of the billet. The number and position of new nucleated grains is chosen randomly from the microstructure calculation units (CA cells or PA points), according to the equation
dn = d ( ∆T ) ς
nmax,ς 2π ∆Tσ ,ς
∆T − ∆Tmax,ς exp − ∆Tσ ,ς 2
2
ς:=s,b,
(1)
where ∆Tmax,s , ∆Tmax,b , ∆Tσ ,s , ∆Tσ ,b , nmax,b and nmax,s represent the mean nucleation undercooling at the surface area, the mean nucleation under-cooling in the bulk area, the standard deviation of the temperature at the surface area, the standard deviation of the temperature at the bulk area, and the maximum density of nuclei that can form in the melt from the surface and bulk, respectively. 2.2
Growth model
The growth velocity V is determined from the KGT [15] model. It is calculated through the following quadratic form:
V 2 A +V B + C = 0,
(2)
where the coefficients A , B and C are modeled as
= A
mC0 (1 − k0 )ξ c π 2Γ = = ,B ,C G , 2 2 D [1 − (1 − ko )Iv(Pe) ] Pe D
(3)
with
= Iv ( Pe ) exp(Pe)erfc
(
Pe
)
= π Pe, ξ c
π 2Γ
RV , = , Pe 2D k0 Pe
(4)
where Γ , D , m , C0 , k0 , Pe, Iv(Pe) and G are Gibbs-Thomson coefficient, diffusion coefficient in liquid, slope of the liquidus line, initial concentration of carbon, partition coefficient, Peclet number for solute diffusion, Ivantsov function, and temperature gradient, respectively. The temperature gradient G at the surface has little influence on the growth velocity and it was set to G = 0 [12] which reduces the equation (2) as follows: V = − B / A . The under-cooling in front of the dendrite tip was solved as follows [15]:
2Γ 1 , = ∆T mC0 1 − + 1 − (1 − k0 ) Iv(Pe) r
(5)
with the dendrite tip radius r , expressed as
r= 3.
ΓD 1 − (1 − k0 ) Iv(Pe) −mV (1 − k0 )C0
.
(6)
CELLULAR AUTOMATA AND POINT AUTOMATA SOLUTION PROCEDURES
Cellular automata discretisation is based on polygons and transition rules between polygons. In the vast majority of all previous science and engineering applications of CA, a regular (in 2 dimensions rectangular or hexagonal) cell structure has been used. Point
automata discretisation is based on points and transition rules between points. Basic concept of PA discretisation is to distribute nodes (not cells) randomly in the domain, which leads to different distances between the nodes and different neighborhood configurations for each of them. Schematics of the CA discretisation and neighborhood configuration is depicted in Fig. 2 and a related schematics of PA discretisation and neighborhood configuration is depicted in Fig. 3. 3.1
Numerical solution of microstructure equations in CA approach
Conventional CA discretisation is generated first and a set of possible neighborhood configurations is determined. Process starts with nucleation where the following conditions need to be checked: appropriate temperature ∆T in the micro cell and the probability condition. During each time step all cells are assigned a random number between ( 0 < rand < 1 ) and a random computational angle from −45 < α < 45 . The transformation from liquid to solid will occur only when rand < p where = p exp ( ∆T − ∆Tmax ) /
(
)
2
2∆Tσ .
Once a cell is nucleated it grows with a preferential direction corresponding to its assigned crystallographic orientation and with respect to the heat flow. Depending on the randomly chosen angle, the neighborhood configuration [8] is chosen (see Fig. 2). The solid phase grows from nucleated cell with different randomly chosen configurations. The growth velocity is calculated according to the KGT model. For all „neighbors” of the treated nucleus, the ratio d is checked by using the formula
d = l ( t ) / aθ ,
t
l (t ) = a tan 2 θ + 1 ∫ V ( ∆T ) dt , aθ =
(7,8)
t0
where t , to , a , θ , and l (t ) are the actual time, initial time, the size of the cell, the crystallographic angle, and the length between the centre of the reference cell and its neighborhood cell, respectively. If a neighbor is one of the four nearest east, north, west, and south neighbours aθ = a , but if a neighbor is a corner neighbor then aθ = a 2 . When d ≥ 1 , the growth front of the solid reference cell can touch the centre of the neighboring cell and then this cell transforms its state from liquid to solid. It is assumed that the growth is not allowed to take place for more than half of micro cell during each time step. 3.2
Numerical solution of microstructure equation in PA approach
Random PA discretisation is generated first. This can be achieved in practices as a random selection of points from the centers of CA cells. As before, process starts with nucleation stage. The position of new grains is chosen randomly from all irregularly spaced points. The nucleation conditions need to be checked [7] as in the CA case. Once a point nucleates it grows with respect to the heat flow and with respect to the ‘neighborhood’ configuration which is now associated with the position of the neighboring points which fall into a circle [3,16] with radius R (see Fig. 3). It means that each point can contain different number and position of the neighbors, which gives various possibilities of neighborhoods. The growth velocity is calculated according to the KGT model. For all neighbors of the treated point, general criterion d is checked by using the formula:
d = l ( t ) / ai ,
l= (t )
t
∫ V ( ∆T ) dt ,
(9,10)
t0
where ai < R represent different lengths from the central to the random points in the circular neighborhood. When d ≥ 1 , the growing solid touches the centre of the neighboring point and this point transforms its state from liquid to solid. 4.
NUMERICAL EXAMPLE AND DISCUSSION OF THE RESULTS
The parameters of the physical model used in PA and CA are exactly the same as in [7]. The regular CA cell size is 200 µ m . The irregular CA nodes have been generated from the regular CA cell size 600 µ m by randomly taking away certain percentage (90 % or 70 %) of the points. The neighborhood configuration of the PA method has been chosen to contain points within circle with radius R (1.2*10-3 m or 3.0*10-3 m) centered around the reference
point. The macroscopic as well as the microscopic models were both coded in Fortran. The calculations in Fig. 4, 5, 6, 7, 8-left, 8-right took 7, 2, 8, 1, 7 and 5 hours to complete.
R R
R R
E
D
G
B A
F
C Rneigborhood
Fig. 2: Scheme of Nastac’s neighborhood configuration [8] for the conventional CA grid
Fig. 3: Scheme of Janssens’s neighborhood configuration [3,16] for the PA structure
Five cases are shown in the paper, depicted in Figures 4, 5, 6, 7 and 8. It is noticed that reducing the number of micro cells which take part in the calculations, the central (equiaxed) zone becomes larger, while the columnar zone is seen only slightly. This can be modified by changing the radius of the neighborhood. Larger the value of R is chosen the wider columnar forms can be observed (Fig. 5). The maximum radius should be kept around R = 3.6*10-3 m, otherwise the columnar structures become distorted (waved). The calculation time grows with smaller radius. Process always starts with nucleation first, followed by a growth stage. Each new grain can start to grow only if the two conditions (temperature and probability) are satisfied. Neighborhoods with a larger number of points have higher probability that at least one of the points will nucleate as well as a higher probability that in the growth process not only one of the neighboring points will be converted to solid. It turns out that by using PA some points might not take part in the process. To avoid this problem, an extra procedure is added, which checks the position of the possible ‘left-out’ nodes which are after identification converted to solid. In our previous [6], conventional CA approach, a sensitivity study of the input parameters was discussed. As a result of this study (Fig. 8-left), a perfect fit of the CA parameters to the experimentally observed microstructure (Fig. 8-center) of a billet of dimension 140 x 140 mm and steel grade 51CrMoV4 was found. We add the PA results in this study (Fig. 8-right). The parameters are the same as in Fig. 4. It can be seen that the two different methods give similar results. 5.
CONCLUSIONS
In this paper, a new PA approach has been demonstrated for prediction of the grain structure which occurs during the CC of steel. PA method offers a simple and powerful approach of cellular simulations. It was shown that both methods are able to qualitatively and quantitatively model a diverse range of solidification phenomena in almost the same calculation time. PA method offers an attractive alternative to classical CA method, because of its flexibility of node density and neighborhood definition. The density of the nodes can in principle vary across the domain of interest and the neighborhood can be defined in a flexible way.
Fig. 4: Simulated grain structure of the billet ECT and CET transformations by PA method,
Fig. 5: Simulated grain structure of the billet ECT and CET transformations by PA method,
R = 1.2*10-3 m, density 90 % of PA. grid
R = 3.0*10-3 m, density 90 % of PA grid
Fig. 6: Simulated grain structure of the billet ECT and CET transformations by PA method,
Fig. 7: Simulated grain structure of the billet ECT and CET transformations by PA method,
R = 1.2*10-3 m, density 70 % of the PA grid
R = 3.0*10-3 m, density 70 % of PA grid
Fig. 8: Center: observed microstructure of charge SS46352, 51CrMoV4 (Baumann print) and simulated results: left: conventional CA method, right: PA method (with parameters from the Fig. 4). Black circle represents approximate position of CET
The new approach has thus theoretical advantages of allowing a more proper and versatile modeling of ECT and CET transformations. Very promising and interesting results
according to the various neighborhood configurations and density of points have been shown. It was also shown that PA method gives compatible results with the conventional CA method when using the same nucleation and growth physics. 6.
ACKNOWLEDGMENT
The first author would like to thank the European Marie Curie Research Training Network INSPIRE for position to study and research at the University of Nova Gorica, Slovenia. Second author wishes to acknowledge the Slovenian Research Agency and Štore - Steel company for funding in the framework of the project L9-9508 Simulation of microstructure for steels with topmost quality. References [1] Flache A., Hegselmann R.: Do irregular grid make a difference? Relaxing the Spatial Regularity in Cellular Models of Social Dynamics, Journal of Artificial Societies and Social Simulation (2001) [2] Janssens K.G.F., Raabe D., Roters F., Barlat F., Chen L.-Q.: Irregular Cellular Automata Modeling of Grain Growth, Continuum Scale Simulation of Engineering Materials (2004), pp. 297-308 [3] Janssens K.G.F.: Random grid, three-dimensional, space-time coupled cellular automata for the simulation of recrystallization and grain growth, Modelling and Simulation in Materials Science and Engineering (2003), pp.157-171 [4] Janssens K. G. F., Raabe D., Kozeschnik E., Miodownik M., Nestler B.: An Introduction to Microstructure Evolution, Computational Materials Engineering, Elsevier (2007) [5] Lee K.Y, Hong C.P.: Stochastic modeling of solidification grain structure of Al-Cu crystalline Ribbons in Planar Flow Casting, ISIJ International (1997), pp. 38-46 [6] Lorbiecka A., Šarler B.: A sensitivity study of grain growth model for prediction of ECT and CET transformations in continuous casting of steel, Materials Science Forum (2008), in print [7] Lorbiecka A., Vertnik R., Gjerkeš H., Šarler B., Cesar J., Manojlović G., Senčič B.: Numerical modelling of the microstructure formation in continuous casting of steel, CMC (2009), pp. 195-208 [8] Nastac L.: Modeling and Simulation of Microstructure Evolution in Solidifying Alloys, Kluwer Academic Publishers USA (2004) [9] Rappaz M., Bellet M., Deville M.: Numerical Modelling in Materials Science and Engineering, Springer Series in Computational Mechanics, Springer, Germany (2003) [10] Šarler B., Vertnik R.: Meshfree local radial basis function collocation method for diffusion problems. Computers and Mathematics with Application (2006), pp. 1269-1282 [11] Šarler B., Vertnik R., Šaletić S., Manojlovič G., Cesar J.: Application of continuous casting simulation at Štore Steel. Berg- und Hüettenmäennische. Monatsheft (2005), pp.300-306 [12] Kurz W., Fisher D.J.: Fundamentals of solidification, Trans. Tech. Publications Ltd, Switzerland (1998) [13] Thieme - Marti S.: Dynamische Zellulaere Automaten, Internal Note Ifu, ETH Zurich (1999) [14] Xu Q.Y., Liu B.C.: Modeling of as-cast microstructure of Al alloy with a modified cellular automaton method, Materials Transactions (2001), pp. 2316-2321 [15] Yamazaki M., Natsume Y., Harada H., Ohsasa K.: Numerical simulation of solidification structure formation during continuous casting in Fe-0.7mass%C alloy using cellular automaton method, ISIJ International (2006), pp. 903-908 [16] Yazdipour N., Davies C., Hodgson P.: Microstructural Modeling of Dynamic Recrystallization Using Irregular Cellular Automata, Computational Materials Science, Elsevier BV, Netherlands (2008), pp. 566-576
